Problem: Let $n = 2^4 \cdot 3^5 \cdot 4^6\cdot 6^7$. How many natural-number factors does $n$ have?
Answer: Prime factorizing gives $n = 2^{23} \cdot 3^{12}$. Since any positive factor of $n$ must be of the form $2^a \cdot 3^b$ where $0 \le a \le 23$ and $0 \le b \le 12$, there are $(23+1)(12+1) = 24 \cdot 13 = \boxed{312}$.